To find the length of the edge of the square base of the Pyramid, we can use the Pythagorean theorem. The pyramid forms a right triangle when we draw a line from the apex (top) of the pyramid to the center of the base and then down to one of the corners of the base.
Let:
- \( h \) be the height of the pyramid (321 ft),
- \( l \) be the slant height of the pyramid (436.5 ft),
- \( a \) be the length of one edge of the square base, and
- \( \frac{a}{2} \) be the distance from the center of the base to one of its corners.
Using the Pythagorean theorem:
\[ l^2 = h^2 + \left(\frac{a}{2}\right)^2 \]
Substituting in the known values:
\[ (436.5)^2 = (321)^2 + \left(\frac{a}{2}\right)^2 \]
Calculating \( (436.5)^2 \) and \( (321)^2 \):
\[ 436.5^2 = 190,372.25 \] \[ 321^2 = 103,041 \]
Now, substituting these values into the equation:
\[ 190,372.25 = 103,041 + \left(\frac{a}{2}\right)^2 \]
Subtract \( 103,041 \) from both sides:
\[ 190,372.25 - 103,041 = \left(\frac{a}{2}\right)^2 \] \[ 87,331.25 = \left(\frac{a}{2}\right)^2 \]
Taking the square root of both sides to find \( \frac{a}{2} \):
\[ \frac{a}{2} = \sqrt{87,331.25} \] \[ \frac{a}{2} ≈ 295.5 \]
Now, we multiply by 2 to find \( a \):
\[ a ≈ 2 \times 295.5 ≈ 591.0 \]
Thus, the length of the edge of the square base is approximately 591.0 ft.
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